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control problem with first-order state constraints
Joseph Frederic Bonnans, Adriano Festa
To cite this version:
Joseph Frederic Bonnans, Adriano Festa. Error estimates for the Euler discretization of an optimal
control problem with first-order state constraints. SIAM Journal on Numerical Analysis, Society for
Industrial and Applied Mathematics, 2017, 55 (2), pp.445–471. �hal-01093229v2�
OF AN OPTIMAL CONTROL PROBLEM
WITH FIRST-ORDER STATE CONSTRAINTS ∗
J. FR ´ED ´ERIC BONNANS† AND ADRIANO FESTA‡
Abstract. We study the error introduced in the solution of an optimal control problem with first order state constraints, for which the trajectories are approximated with a classical Euler scheme.
We obtain order one approximation results in the L∞norm (as opposed to the order 2/3 obtained
in the literature). We assume either a strong second order optimality condition, or a weaker one in the case where the state constraint is scalar, satisfies some hypotheses for junction points, and the time step is constant.
Our technique is based on some homotopy path of discrete optimal control problems that we study using perturbation analysis of nonlinear programming problems.
Key words. Optimal control, nonlinear systems, state constraints, Euler discretization, rate of convergence.
AMS subject classifications. 49M25, 65L10, 65L70, 65K10.
1. Introduction, discussion of literature. Numerical methods for the reso-lution of an optimal control problem are based on a finite dimensional approximation, generally obtained through a discretization of the trajectory and a piecewise constant or polynomial control. Obtaining error estimates for such approximations is obviously an important issue.
The first works related appeared in the 1970s; they dealt with convergence of a discrete optimal control solution (see e.g. [9], [10], and [23]). Other results of convergence, provided with modern variational techniques, are also [26]; a survey of the results in this area is [11].
In this paper we will focus on the case of pure state constraints, a case which presents some special difficulties. In particular it is known that when the constraint qualification (see [14]) holds and the Lagrangian verifies a local condition of coerciv-ity, the discrete problem obtained with an Euler scheme has a solution, for a suffi-ciently fine mesh, and the corresponding Lagrange multipliers are at distance O(¯h), in the L2 norm, where ¯h is the maximal discretization step, from the continuous
solution/multiplier. This important result is due to [13].
The choice of the norm is a delicate point: through the Legendre-Clebsch condi-tion we can get typically an estimacondi-tion for our variables in a L2 norm which settles
badly with the pure state constrained problem. Such problem naturally requires esti-mations in the L∞ norm. This is the so called “two-norm discrepancy” [21].
Another sensible matter is that the cost function does not necessarily have deriva-tives in L2. This suggests to work in a non linear space of Lipschitz continuous
func-tions with bounded Lipschitz constants. In this setting the L2 convergence implies
L∞ convergence. This is the way proposed in [13] to obtain a convergence result in the L∞ norm. This reference obtains an error bound, of order O(¯h2/3).
We assume either (i) a strong second order optimality condition, similar to the one in [13], (but we allow a variable time step, whereas the time step was constant in
∗This work was partially supported by the European Union under the 7th Framework Programme
FP7-PEOPLE-2010-ITN SADCO, Sensitivity Analysis for Deterministic Controller Design.
†Inria and CMAP, Ecole Polytechnique, 91128 Palaiseau, France (frederic.bonnans@inria.fr).
‡RICAM, Austrian Academy of Science, 4040 Linz, Austria (afesta@oeaw.ac.at).
that reference), or (ii) a weaker second order optimality condition, in the case when the state constraint is scalar, structural hypotheses on arcs and junction points, and the time step is constant (the precise statements of these hypotheses are in section 2.5).
In this second case our hypotheses can be motivated in the following way. They allow to obtain the stability of the extremals (of the continuous problem) under a small perturbation, see [3]. We obtain a similar result for the discretized problem. By contrast, for a vector state constraint we are not aware of such stability results, even in the continuous case. This suggests that it might be not easy to obtain the stability of the extremals after discretization without a strong second order optimality condition. This is an interesting open question that we leave for future work, as well as the case of higher order state constraints.
1.1. Structure of the paper. In Section 2 we introduce the problem and the assumptions adopted in the paper, and we state our main result (Theorem 2.6), i.e., a O(¯h) error estimate for the control, state, costate and multiplier. A key role in our construction is played by an homotopy path introduced in Section 3. The path links the continuous problem to the discrete one, involving the control, state, costate and multipliers, and creating a class of auxiliary problems. Through the study of the regularity of each auxiliary problem (Section 4) and checking that, under appropri-ate hypotheses, the application obtained (homotopy path) has bounded directional derivatives (in a sense clarified in the devoted section 5), we can establish the an-nounced convergence estimates for the discrete problem. More precisely, due to some coercivity properties of the Hessian of the Lagrangian, we first obtain a bound in the L2 norm from which respective estimates in the L∞ norm easily follow. In this
analysis it is used the fact that the state constraint is of first order. Section 6 is dedicated to a simple numerical test. The numerical results are in accordance with our theoretical result and they confirms the tightness of the estimate. An appendix is devoted to the analysis of hypothesis (A5).
1.2. Notations. By Rn we denote the n dimensional Euclidean space. Its dual
(whose elements are row vectors) is denoted by Rn∗. By ∇, ∇
u, etc. we denote
the gradient or partial gradient w.r.t. u, who are column vectors, by contrast to the derivatives denoted by e.g. Dg(x) or g0(x) depending on the context, which are identified to row vectors if g is scalar valued. The Lagrange multipliers, including costate variables, are considered as dual elements and are represented by row vectors. By C([0, T ] we denote the space of real continuous functions over [0, T ], endowed with the supremum norm. It is known that its topological dual can be identified with the space M[0, T ] of regular, finite Borel measures over [0, T ]. Let BV ([0, T ]) denote the space of bounded variation functions over [0, T ], and let BVT([0, T ]) be
the subspace of such functions with value 0 at time T . Any continuous linear form on C([0, T ] is of the form f 7→RT
0 f (t)dµ(t), with µ ∈ BVT([0, T ]).
2. The continuous problem and its discretization. We consider the follow-ing pure state constrained optimal control problem
(P)
Minimize φ(y(T )); subject to
˙
y(t) = f (u(t), y(t)), for a.a. t ∈ [0, T ]; y(0) = y0;
gi(y(t)) ≤ 0, t ∈ [0, T ], i = 1, ..., r,
where the initial condition y0 ∈ Rn, the control u(t) and the state y(t) belong to
the spaces U := L∞(0, T ; Rm) and Y := W1,∞
(0, T ; Rn), resp., and g
i is the i-th
component of the vector g. Moreover we assume: (A0) The mappings φ : Rn → R, f : Rm× Rn→ Rn
, and g : Rn→ Rrare of class
C2with locally Lipschitz second order continuous derivatives.
In addition, the initial condition y0∈ Rn satisfies gi(y0) < 0, i = 1, ..., r.
A trajectory of (P) is an element (u, y) of U × Y solution of the state equation (2.1). We say that (˜u, ˜y) is a local solution of (P), if it minimizes φ(·) over the set of feasible trajectories (u, y) satisfying ku − ˜uk∞≤ δ for some δ > 0. We assume that
(A1) The nominal trajectory (¯u, ¯y) is a local solution of (P) in U × Y, and ¯u is a continuous function of time.
The first order time derivative of the state constraint is the function g(1): Rm× Rn
→ Rr, (u, y) → g0(y)f (u, y). (2.2)
Note that g(1)(¯u, ¯y) is the time derivative of g(¯y) along a trajectory.
Denote the set of active constraints at time t by A(t) := {i = 1, ..., r | gi(¯y(t)) = 0}.
We say that the trajectory (¯u, ¯y) has regular first order state constraints if the following holds:
(A2) There exists αg > 0 such that, for all t ∈ [0, T ] and λ ∈ Rr∗ verifying λi= 0
if i 6∈ A(t), the following holds:
|λ| ≤ αg X i∈A(t) λi∇ug (1) i (¯u(t), ¯y(t)) .
The Hamiltonian function H : Rm
× Rn
× Rn∗
→ R is defined by: H[p](u, y) := pf (u, y).
With this classical notation we view the Hamiltonian as a function of (u, y), parame-terized by p, so that e.g. DH[p](u, y) denote the derivative of the Hamiltonian w.r.t. (u, y).
For i = 1 to r, we define the contact set for the i-th constraint by Ii:= {t ∈ [0, T ]; gi(¯y(t)) = 0}.
We say also that the i-constraint is active at time t, if t ∈ Ii; otherwise the constraint
will be inactive. A maximal open interval (a, b) of Ii (resp. of [0, T ] \ Ii) is called
boundary arc (resp. interior arc). The left and right endpoints of a boundary arc are called entry and exit points, respectively. We call junction points the union of entry and exit points.
2.1. Optimality conditions. We next introduce Pontryagin extremal in a qual-ified form, which is convenient in view of hypothesis (A2).
Definition 2.1. A trajectory (¯u, ¯y) is a regular Pontryagin extremal of (P), if there exist ¯η ∈ BVT([0, T ], Rr) and p ∈ BV ([0, T ], Rn∗), such that:
˙¯
y(t) = f (¯u(t), ¯y(t)) a.e. on [0, T ], y(0) = y¯ 0, (2.3)
− d¯p(t) = ¯p(t)fy(¯u(t), ¯y(t))dt + r X i=1 g0i(¯y(t))d¯ηi(t), p(T ) = φ¯ 0(¯y(T )), (2.4) ¯ u(t) ∈ argmin ˜ u∈Rm {¯p(t)f (˜u, ¯y(t))} a.e. on [0, T ], (2.5) 0 ≥ gi(¯y(t)), d¯ηi≥ 0, Z T 0 gi(¯y(t))d¯ηi(t) = 0, i = 1, ..., r. (2.6)
We call ¯η a Pontryagin multiplier, and p a costate. Observe that condition (2.5) is equivalent to the Hamiltonian inequality
H[¯p(t)](¯u(t), ¯y(t)) ≤ H[¯p(t)](u, ¯y(t)), for all u ∈ Rm, a.e. on [0, T ]. (2.7) A trajectory (u, y) is a stationary point of (P), if there exist ¯η ∈ BVT([0, T ], Rr) and
¯
p ∈ BV ([0, T ]; Rn∗) such that (2.3), (2.4), (2.6) hold, as well as
0 = Hu[¯p(t)](¯u(t), ¯y(t)) = ¯p(t)fu(¯u(t), ¯y(t)) for a.a. t ∈ [0, T ].
Then we call ¯η a Lagrange multiplier. Obviously, a regular Pontryagin extremal is also a stationary point, and the converse holds if the Hamiltonian H is a convex function of the control for a.a. time.
The linearized state equation at (¯u, ¯y) is, for v ∈ L2(0, T )m:
˙
z(t) = f0(¯u(t), ¯y(t))(v(t), z(t)); z(0) = 0, (2.8) and we denote its solution by z[v].
Theorem 2.2. Any qualified solution of (P) is a regular Pontryagin extremal. Proof. It is known that a solution of the problem satisfies Pontryagin’s principle in unqualified form, see e.g. [27]. On the other hand, by (A2), (¯u, ¯y) satisfies the following constraints qualification [24] (cf. also [6]): there exists εQ> 0 and v]∈ L∞
such that z]= z[v]] satisfies
gi(¯y(t)) + gi0[t]z
](t) < −ε
Q< 0, t ∈ [0, T ], i = 1, ..., r. (2.9)
But while the above qualification condition only guarantees the fact that the set of Lagrange multipliers is nonempty and bounded, (A2) implies the uniqueness of the Lagrange multiplier, see [5].
2.2. A key result. The next assumption is quite common in these problems and it plays a crucial role in the analysis. We assume that problem (P) has a local solution (¯u, ¯y), with associated multipliers ¯p and ¯η satisfying the following condition
(A3) (Strengthened Legendre-Clebsch condition) There exists α > 0 such that Huu[¯p(t)](¯u(t), ¯y(t)) ≥ α, for a.a. t ∈ [0, T ]. (2.10)
We recall that the continuity of the control was stated in (A1). A sufficient condition for the continuity of the control, stronger than (A3), is (see [4, Thm. 2]) the uniform
strong convexity of the Hamiltonian w.r.t. the control variable, i.e. there exists α > 0, such that
Huu[¯p(t)](ˆu, ¯y(t)) ≥ α, for all ˆu ∈ Rmand t ∈ [0, T ]. (2.11)
Observe that, when ¯u and ¯η are Lipschitz continuous, denoting by ν(t) the density of ¯
η, the costate equation can be written in the form
− ˙¯p(t) = ¯p(t)fy(¯u(t), ¯y(t)) + r X i=1 νi(t)gi0(¯y(t)) a.e. on (0, T ); p(t) = φ¯ 0(¯y(T )). (2.12) Lemma 2.3. Let (A0)-(A3) hold. Then both ¯u and ¯η are Lipschitz continuous, and (2.12) holds.
Proof. The result is proved in [14, Thm 4.2] in the case of a convex problem, using a Lemma on ‘compatible pairs’. It was generalized in [1], using the same Lemma, to non convex problems with state constraints of any order.
In the rest of the paper we assume as standing hypothesis (A0)-(A3).
2.3. Second Order Conditions and Alternative Formulation. Let us first recall some theoretical issues about second-order conditions. We introduce the lin-earized control and state space V := L2(0, T ) and Z := H1(0, T ; Rn), resp. We use also the notations
H[t] := H[¯p(t)](¯u(t), ¯y(t)); g[t] := g(¯y(t)), f [t] := f (¯u(t), ¯y(t));
as well as for their partial derivatives, e.g. Hu[t] := Hu[¯p(t)](¯u(t), ¯y(t)), and other
functions. Let us define the quadratic form over V × Z, where z = z[v]:
Ω(v) := Z T 0 Hyy[t](v(t), z(t))2+ r X i=1 νi(t)g 00 i[t](z(t)) 2 ! dt + φ00(¯y(T ))(z(T ))2, (2.13)
and the set C(¯u) of strict critical directions is defined as those v ∈ V such that, for z = z[v]:
˙
z = fu(¯u, ¯y)v + fy(¯u, ¯y)z on [0, T ]; z(0) = 0, (2.14)
gi0(¯y(t))z(t) = 0, t ∈ Ii (2.15)
φ0(¯y(T ))z(T ) = 0. (2.16)
Note that in the last relation we write an equality instead of an inequality since this is known to be equivalent for qualified solutions, which will be the case thanks to assumption (A2).
Let us next recall the alternative formulation of the optimality conditions, due to [8] and [17], and put on a sound mathematical basis by [20]. (See also [3]). The alternative Hamiltonian H : Rm× Rn
× Rn∗
× Rr∗
→ R is defined by: ˜
H[p1, ¯η1](u, y) := p1f (u, y) + ¯η1g(1)(u, y). (2.17) Now define the alternative costate and multiplier of the state constraint:
p1(t) := ¯p(t) + r X i=1 ¯ ηi(t)gi0(¯y(t)); η¯ 1 (t) := −¯η(t), t ∈ (0, T ).
It is easily checked that
− ˙p1(t) = ˜Hy[p1(t), ¯η1(t)](¯u(t), ¯y(t)) a.e. on (0, T ); p1(T ) = φ0(¯y(T )). (2.18)
At the same time, for any u ∈ R, we have that ˜
H[p1(t), ¯η1(t)](u, ¯y(t)) = (p1(t) + ¯η1(t)g0(¯y(t)))f (u, ¯y(t)) = H[¯p(t)](u, ¯y(t)). Consequently the property of stationarity or minimization of the Hamiltonian w.r.t. the control holds for the original Hamiltonian H if and only if it holds for the alter-native Hamiltonian ˜H.
The corresponding alternative quadratic form, where z = z[v], has the following expression: ˜ Ω(v) := Z T 0 ˜ Hyy[t](v(t), z(t))2dt + φ00(¯y(T ))(z(T ))2.
The form above involves the expression of D2g(1)[t], which is easily checked to be
D2g(1)[t](v, z)2= g(3)[t](f [t], z, z) + g0[t]fyy[t](v, z)2+ 2g00[t](z, fy[t](v, z)). (2.19)
The next Lemma is a variant of some results by Bonnans and Hermant [3], Malanowski and Maurer [19]. We give a short, direct proof in the case of a single constraint for convenience.
Lemma 2.4. We have that ˜Ω(v) = Ω(v), for all v ∈ U . Proof. Using (2.19), we get that
Z T 0 ν(t)g00[t](z(t))2dt = Z T 0 g00[t](z(t))2d¯η(t) = Z T 0 d dt g 00[t](z(t))2 ¯η1(t)dt = Z T 0 g(3)(¯y(t))(f [t], z(t), z(t)) + 2g00[t](z(t), ˙z(t))η¯1(t)dt = Z T 0 D2g(1)[t](v(t), z(t))2− g0[t]fyy[t](v(t), z(t))2 ¯ η1(t)dt.
Eliminating ¯p(t) = p1(t) + ¯η1g0[t] in the expression of Ω(·), we obtain that
Ω(v) = Z T 0 (p1(t) + ¯η1(t)g0[t])fyy[t](v(t), z(t))2+ ν(t)g00[t](z(t))2 dt +φ00(¯y(T ))(z(T ))2 = Z T 0 p1(t)fyy[t](v(t), z(t))2+ ¯η1(t)D2g(1)[t](v(t), z(t))2 dt +φ00(¯y(T ))(z(T ))2, as was to be proved.
2.4. Discrete version. We introduce now the Euler discretization of the opti-mal control problem (2.1). Given some non zero N ∈ N and a collection of positive time steps hk, k = 0 to N − 1, such thatP
N −1 k=0 hk= T , we set tk:= k−1 X i=0 hi, k = 0, . . . , N ; ¯h = max k=0,...,N −1hk,
and we consider the discretized problem (Pd)
Minimize φ(yN); subject to
yk+1= yk+ hkf (uk, yk), for k = 0, . . . , N − 1;
y0= y0;
g(yk) ≤ 0 for k = 1, . . . , N.
(2.20)
We denote by UN the space of discrete control variable. The associated Lagrangian
function (with a proper scaling of the state constraint) is
φ(yN) + N −1 X k=0 pk+1(yk+ hkf (uk, yk) − yk+1) + N X k=1 hkνkg(yk), where νkg(yk) =P r
i=1νk,igi(yk). The first-order optimality conditions (in qualified
form), for this finite dimensional optimization problem with finitely many equalities and inequalities, are
pk = pk+1+ hkpk+1fy(uk, yk) + hkνkg0(yk), k = 0, . . . , N − 1,
pN = φ0(yN) + hNνNg0(yN),
0 = Hu[pk+1](uk, yk), k = 0, . . . , N − 1,
gi(yk) ≤ 0, νk,i≥ 0; νk,igi(yk) = 0, i = 1, ..., r, k = 0, . . . , N.
(2.21)
Analogously to the continuous case we define also the ’integrated’ multiplier of the normal and the alternative formulation
ηk:= − N X j=k hkνk, η := −η,¯ (2.22) so that hkνk = ηk+1− ηk= ¯ηk− ¯ηk+1, for k = 0, . . . , N .
Definition 2.5. We say that the discretization step is constant if
h0= h1= · · · = hN −1. (2.23)
2.5. Main result. As mentioned before, our results hold in two different cases. We need to preserve the coercivity of the Hessian of the Lagrangian over some sub-space; this can be stated as hypothesis, as in [13] or obtained under structure hy-potheses for a scalar state constraints. So we will assume that one of the following assumptions hold:
(A4) There exists a constant α > 0 such that
Ω(v) ≥ α Z T
0
|v(t)|2dt, for all v ∈ U
and all discrete steps are of the same order, i.e. max
k (hk/hk−1+ hk−1/hk) = O(1). (2.24)
The condition on Ω is known to be a sufficient condition for local optimality in U . This follows from [22, Th. 5.6].
(A5) (scalar constraint and finite structure) Assume that r = 1, the discretization step is constant, the set I is a finite union of boundary arcs, the density ν is uniformly positive over the boundary arcs, and there exists a constant α > 0 such that
Ω(v) ≥ α Z T
0
|v(t)|2dt, whenever v ∈ C(¯u). (2.25)
The set of strict critical directions C(¯u) is defined in (2.14)-(2.16).
This condition on Ω is known to be a sufficient condition for local optimality in U , see [3, Th. 3.8].
Note that (A5) excludes touch points, i.e., isolated elements of Ii, for i = 1, . . . , r.
We denote by Ibthe union of boundary arcs, of the form Ib := ∪Nb
j=1[Tjen, Tjex] where
we have ordered the set of entry points Ten:= {Ten
1 < · · · < TNenb}, and similarly for
the set Tex of exit points.
Theorem 2.6. Let assumptions (A0)-(A3) hold as well as either (A4) or (A5). Then the discrete optimal control problem (Pd) has a local solution (uh, yh) with
as-sociated multipliers (ph, ηh), such that kyh− ¯yk
∞+ kuh− ¯uk∞+ kph− ¯pk∞+ kηh− ¯ηk∞= O(¯h). (2.26)
The rest of the paper will be dedicated to the proof of this result; for that purpose we need to introduce a special auxiliary problem.
3. Homotopy path. We consider a family (Pθ) of perturbed discrete optimiza-tion problems, parametrized by θ ∈ [0, 1]. The definioptimiza-tion is done such that for θ = 0, the problem reduces to (Pd). As we will see in the next Section, a certain sampling of
the solution of the continuous problem (P) happens to be a solution of the optimality system of (P1). The perturbed optimal control problem is, for some perturbations terms δp, δu, δy, δg, to be defined later, such that kδ·k
∞= O(1), (Pθ) Minimize φ(yθ N) + θ PN −1 k=0 h2k(δ p kykθ+ δukuθk); subject to yθ k+1= yθk+ hkf (uθk, yθk) + θh2kδ y k, for k = 0, . . . , N − 1; gi(yθk) ≤ θh2kδ g k, for k = 1, . . . , N, yθ 0= y0, i = 1, ..., r. (3.1)
The corresponding optimality system is pθ k = p θ k+1+ hkpθk+1fy(uθk, y θ k) + hkνkθg0(y θ k) + θh 2 kδ p k, pθ N = φ0(yNθ) + hNνNθg0(yNθ), 0 = Hu[pθk+1](uθk, ykθ) + θhkδuk, (3.2)
for k = 1 to N − 1, with the complementarity conditions, i = 1, ..., r, gi(yθk) − θh2kδ g k ≤ 0, νk,iθ ≥ 0, νk,iθ (gi(ykθ) − θh2kδ g k) = 0, k = 1, . . . , N. (3.3) Let us set fk := (uθk, ykθ); Hk:= H[pθk+1](uθk, yθk) (3.4)
For future reference we note that the linearization of the costate equation is, denoting by (vθ, zθ, qθ, δνθ) the linearizations of (uθ, yθ, pθ, νθ): qθ k= q θ k+1+ hkqθk+1f k y + hk(vkθ) THk uy+ hk(zkθ) THk yy +hkνkθ(zkθ)Tg00(ykθ) + hkδνkθg0(ykθ) − θh2kδ p k, qθ N = (z θ N) Tφ00(yθ N) + hNνNθ(z θ N) Tg00(yθ N) + hNδνNθg 0(yθ N). (3.5)
The corresponding approximation of ¯η is: (see (2.22))
ηθk:= −
N
X
j=k
hjνjθ, k = 0, . . . , N. (3.6)
The sampling of the continuous solution and the associated multipliers of the original problem (2.1) are defined by
ˆ uk := ¯u(tk), yˆk:= ¯y(tk), k = 0, . . . , N − 1, ˆ pk := ¯p(tk), νˆk:= Z tk+1 tk ν(t)dt, k = 1, . . . , N, (3.7)
and accordingly we can define
ˆ ηk:= η(tk+1) = N X j=k hkνˆk, k = 1, . . . , N. For θ = 1 we define uθ
k and the associated state and multipliers by
u1k= ˆuk, y1k= ˆyk, p1k = ˆpk, η1k= ˆηk, for k = 1, ..., N . (3.8)
We next give to the perturbation terms (δuk, δky, δkp, δkg) the unique value such that the above sampling is solution of the discretized problem for θ = 1.
Lemma 3.1. We have that kδyk
∞+ kδuk∞+ kδpk∞+ kδgk∞= O(1). (3.9)
Proof. Since u is Lipschitz continuous by Lemma 2.3, t → g(y(t)) has a.e. a bounded second derivative. Therefore, if νk 6= 0 there exists some c > 0 such that
g(y(t)) ≥ −c¯h2for all t ∈ [t
k, tk+1], so that kδgk∞= O(1).
Next, if w : [0, T ] → R is C1 with a Lipschitz continuous derivative of constant
L, then by a first order Taylor expansion, we have that |w(t + h) − w(t) − w0(t)h| ≤12Lh2.
By Lemma 2.3, the control is Lipschitz, and therefore, so does ˙y(t); we deduce that kδyk
∞= O(1). For the costate equation, we have that
¯ p(tk+1) = ¯p(tk) − Z tk+1 tk ¯ p(t)fy[t]dt − Z tk+1 tk ν(t)g0[t]dt.
Now since ¯u, ¯y, and ¯p are Lipschitz, Z tk+1 tk ¯ p(t)fy(¯u(t), ¯y(t))dt − Z tk+1 tk ¯ p(tk+1)fy(¯u(tk), ¯y(tk))dt = O(h2k),
and (in the parenthesis below we recognize the expression of νk): Z tk+1 tk ν(t)g0[t]dt − Z tk+1 tk ν(t)dt g0(¯y(tk)) = O(h2k).
It follows that kδpk∞= O(1). Since p is Lipschitz and Hu[p(tk)](¯u(tk), ¯y(tk)) = 0 we
easily deduce that kδuk∞= O(1). The conclusion follows.
In the rest of the paper some of the estimates presented are valid in a special neigh-borhood of the continuous solution. To state them rigorously, we need the following definition: given ε > 0 and θ ∈ [0, 1], we say that a solution Xθ := (uθ, yθ, yθ, ηθ) of the optimality system (3.2) is an ε-neighboring solution if we have that
kuθ− ˆuk
∞+ kyθ− ˆyk∞+ kpθ− ˆpk∞+ kηθ− ˆηk∞≤ ε, (3.10)
and we define
θm:= inf{θ ∈ [0, 1]; (3.2) has an ε-neighbouring solution}.
When θ = 1, the l.h.s. of (3.10) has value 0, and therefore θm is well-defined with
value in [0, 1].
Through the auxiliary structure of the homotopy path problem we are now able to prove the bounds shown in Theorem 2.6. In the sequel we will analyze some technical points. In particular we will use the fact that the solution of (Pθ) is uniformly Lipschitz continuous (the result is contained in Section 4) and that such solution is in a ε-neighborhood of the solution of (P), X1= (¯u, ¯y, ¯p, ¯η); this point will be discussed in Section 5. Note that we define the Lipschitz constant of a function defined for discrete times, as e.g. for uθ, by
Lip(uθ) := max |uk− uk−1|/h
k; k = 0, . . . , N − 1 .
Proof. [of Th. 2.6.] We prove in Section 4 that, if ¯h is small enough, for θ ∈ [θm, 1],
then (uθ, yθ) is uniquely defined, has unique associated multipliers (pθ, ηθ), and setting
Xθ := (uθ, yθ, pθ, ηθ), (see Section 5) θ → Xθ has Lipschitz constant of order ¯h. It
follows that, for a fix ε > 0, kXθ− X1k
∞< ε when ¯h is small enough, which gives a
contradiction if θm> 0. Therefore, X0 is well-defined and kX1− X0k∞ = O(¯h), as
was to be shown.
4. Regularity of the solutions. In this Section we present some regularity results for the solutions of the Homotopy path problem.
Given an ε−neighboring solution Xθ of (Pθ), we will prove the uniqueness and the
uniform Lipschitz continuity of Xθ. In the following, when there is no possibility of
confusion, we drop θ as upper index in the notation for a better readability, keeping the complete notation in the statements of the main propositions. We need to define
Ck1 := pk hk (fu(uk−1, yk) − fu(uk−1, yk−1)) , Ck2 := pk+1fy(uk, yk)fu(uk, yk), ∆uk := hkδku− hk−1δuk−1 = (ˆpk− ˆpk+1)fu(ˆuk, ˆyk) − (ˆpk−1− ˆpk)fu(ˆuk−1, ˆyk−1). C3 k := C 1 k− C 2 k− θhkδ p kfu(uk, yk) + θ∆uk/hk. (4.1)
Observe that, since yk is uniformly Lipschitz we have that:
(i) Ck1= hk−1 hk
Huy[pk](uk−1, yk)f (uk−1, yk−1) + O(¯h); (ii) |Ck3| = O(1). (4.2)
Lemma 4.1. (i) Let Xθ be an ε-neighboring solution of (Pθ). Then there exists cH > 0 such that, if ε > 0 and ¯h are small enough, then
r X j=1 νk,jθ ∇ug (1) j (u θ k, y θ k) = H θ k (uθk− uθ k−1) hk + Ck3, (4.3) where Hθ k satisfies |Hθ k− Huu[pk](uk, yk)| ≤ cHε, k = 0, . . . , N − 1. (4.4)
(ii) Let ε0 > 0. If in addition, the time step is constant, tk belongs to a boundary
arc (ta, tb), and tka+ ε
0 < t
k−1 < tk < tkb− ε
0, then the variation of C3
k along the
homotopy path is of order O(¯h).
Proof. (i) Note that hkδuk = (ˆpk− ˆpk+1)fu(ˆuk, ˆyk). By the optimality condition
(3.2), we have that 0 = Hu[pk+1](uk, yk) − Hu[pk](uk−1, yk−1) + θ∆uk = pk+1fu(uk, yk) − pkfu(uk−1, yk−1) + θ∆uk = (pk+1− pk)fu(uk, yk) + pk[fu(uk, yk) − fu(uk−1, yk−1)] + θ∆uk = (pk+1− pk)fu(uk, yk) + pk[fu(uk, yk) − fu(uk−1, yk)] + hkCk1+ θ∆ u k. (4.5) By the mean-value Theorem, we deduce that
pk(fu(uk, yk) − fu(uk−1, yk)) = Hk(uk− uk−1), where Hk:= pk Z 1 0 fu(uk−1+ σ(uk−1− uk), yk)(uk−1− uk)dσ,
so that (4.4) holds. We conclude by combining (4.5) and the discrete costate equation in (3.2), where ∇ug
(1)
i (uk, yk) = gi0(yk)fu(uk, yk).
(ii) It is easily checked that C1
k and Ck2 satisfy this property, as well as (it is of
order of ¯h) θhkδkpfu(uk, yk). Since the time step is constant, we have that by (4.1)
∆uk/hk = δku− δ u
k−1is of order of ¯h over the interior of a boundary arc. The conclusion
follows. Let us define ∆k,ig := gi(yk+1) − gi(yk) hk −gi(yk) − gi(yk−1) hk−1 . Ξθ k := g 0 i(yk) (f (uk−1, yk) − f (uk−1, yk−1)) +θgi0(yk)(hkδyk− hk−1δyk−1) + 1 2hkg 00 i(yk)f (uk, yk)2 +12hk−1gi00(yk)f (uk−1, yk−1)2 (4.6)
a) the following relation holds
∆k,ig = gi0(yk)fu(yk, uk)(uk− uk−1) + Ξθk+ O(¯h
2) + O(ε|u
k− uk−1|). (4.7)
b) If in addition the time step is constant, then kΞθ− Ξ1k
∞= O(¯hε). (4.8)
Proof. Use
gi(yk+1) − gi(yk) = gi0(yk)(yk+1− yk) +12g00i(yk)(yk+1− yk)2+ O(h3k),
= hkgi0(yk)f (uk, yk)
+12h2k 2θg0(yk)δ y
k+ gi00(yk)f (uk, yk)2 + O(h3k),
gi(yk−1) − gi(yk) = gi0(yk)(yk−1− yk) +12g00i(yk)(yk−1− yk)2+ O(h3k),
= −hk−1gi0(yk)f (uk−1, yk−1) +12h2k−1 −2θg0 i(yk)δ y k−1+ gi00(yk)f (uk−1, yk−1)2 +O(h3 k−1).
We obtain a) by dividing these relations by hkand hk−1respectively, adding them
and observing that
f (uk, yk) − f (uk−1, yk−1) = f (uk, yk) − f (uk−1, yk) + f (uk−1, yk) − f (uk−1, yk−1)
= fu(uk, yk)(uk− uk−1) + (f (uk−1, yk) − f (uk−1, yk−1)) + O(ε|uk− uk−1|).
Since |yk − yk−1| = O(hk−1), point (b) follows using that hk = hk−1 and |δky−
δyk−1| = O(¯h).
Now we are ready to obtain the uniform Lipschitz estimates of the variables of the perturbed problem. A similar result, in the case of a linear quadratic optimal control problem was obtained in [12]. Let us set
wk= νk∇ug(1)(uk, yk) = r X i=1 νk,i∇ug (1) i (uk, yk). (4.9)
Lemma 4.3. We have that wTkHk−1wk = 1 hk wk· (uk− uk−1) + wTHk−1C 3 k, (4.10)
as well as, called Lip(uθ) and Lip(pθ) the Lipschitz constant of uθ and pθ,
Lip(uθ) + Lip(pθ) + kνθk∞= O(1).
Proof. By the Legendre-Clebsch condition (A3), for small enough ε > 0, Hk is
uniformly invertible. Computing the scalar product of both sides of (4.3) by wT kH
−1 k ,
we obtain (4.10).
When νk,i 6= 0 we have that gk,iθ := gi(yk) − θh2kδ g
k,i reaches a local maximum and
therefore 0 ≥ g θ k+1,i− gk,iθ hk −g θ k,i− gθk−1,i hk−1 ,
which amounts to ∆k,ig ≤ θ h 2 k+1δ g k+1,i− h 2 kδ g k,i hk −h 2 kδ g k,i− h 2 k−1δ g k−1,i hk−1 ! ≤ O(¯h). (4.11) Since |C3
k| = O(1) as already noticed, it follows from (4.3) that |uk − uk−1|/hk =
O(|νk| + 1). Now putting it together with (4.7) and (4.11), it follows that
1 hk wk· (uk− uk−1) + wTH−1Ck3≤ 1 hk r X i=1
νk∆k,ig + O(1) + O(|νk|), (4.12)
then for the linear independence of the constraints w.r.t. the control (Assumption (A2), i.e. |wk| ≥ αkνkk), in the l.h.s. of (4.10), we have
α|νk|2≤ O(|νk|) + O(1). (4.13)
By the above display, |νk| = O(1), and by (4.3), uk is uniformly Lipschitz. By (3.2),
so is the discretized costate. 5. Sensitivity analysis.
5.1. Characterization of directional derivatives. In this Section we com-plete the proof of Theorem 2.6 showing that a solution (uθ, yθ) of the perturbed problem (Pθ) is in a L∞ neighborhood of (¯u, ¯y), local solution of the problem (P).
The strategy consists in establishing that the path Xθ:= (uθ, yθ, pθ, ηθ) is Lipschitz,
and then to show that it has directional derivatives δXθ:= (vθ, zθ, qθ, δηθ) satisfying
kδXθk
∞= O(¯h). This will imply that the Lipschitz constant of Xθ(in the L∞norm)
is of order O(¯h). We define (compare to (2.22)) νθ k := η
θ k+1− η
θ
k, k = 0 to N .
The fact that Xθ is Lipschitz is a consequence of Robinson’s theory for strong
regularity [25] and its application to nonlinear programming see e.g. [2, Sec. 5.1]. This theory gives a sufficient condition for Lipschitz stability of the local solution and associated multiplier, provided that we have the (i) linear independence of gradients of active constraints, which by (A2) always holds, and (ii) positivity of the Hessian of the Lagrangian over an extended critical cone, obtained by removing inequality constraints associated with zero components of the multiplier, as well as the condition on the linearization of the cost function. In addition, under these conditions, Jittorntrum’s result [18] states that directional derivatives exist and that they are solution of the following quadratic programming problem
(QP ) Min v 1 2Ω θ(v, z) − θ N −1 X k=0 h2k(δpkzk+ δkuvk) s.t. z = zθ[v] and g0 i(yk)zk = −θh2kδ g k,i k ∈ I i,θ + , s.t. νk,iθ 6= 0, g0i(yk)zk ≤ −θh2kδ g k,i k ∈ I i,θ 0 , s.t. νk,iθ = 0, (5.1)
Here v ∈ VN, zθ[v] ∈ ZN is defined as the unique solution of the linearized state equation
zθk+1= zkθ+ hkfy(uk, yk)(vk, zkθ) − θh2kδ y
k, k = 0, ..., N − 1, i = 1, ..., r, (5.2)
and the set of constraints I+i,θ and I i,θ
0 are defined as the inequality constraints of
problem (Pθ) that are active at yθ, i.e.:
(
I+i,θ := {k = 0, . . . , N ; νθ k,i > 0},
I0i,θ := {k = 0, . . . , N ; νk,iθ = gi(yθk) = 0}.
Finally, the Hessian of Lagrangian of the discretized problem is, with obvious nota-tions, setting Hθ k := pθk+1f (uθk, yθk), and for zθ= zθ[v]: Ωθ(v, z) := N −1 X k=0 hkD2Hkθ(vk, zk)2+ N X k=0 hkνkθD2gkθ(zk)2+ φ00(yNθ)(zN)2. (5.4)
We anticipate a result that will be shown in details later (Corollary 5.6) about the existence and uniqueness of the solution of (QP ):
Proposition 5.1. Let (A0)−(A3) and, either (A4) or (A5) hold. Then problem (QP ) has a unique solution δXθ:= (vθ, zθ, qθ, δηθ).
Let us introduce the following alternative formulation: we underline the analogy with the alternative formulation recalled in Section 2.3. We first define the set of inequality constraints that are active at the solution of (QP ):
Ii,θ := I+i,θ∪ {k ∈ I0i,θ; gi0(y θ k)z
θ
k = 0}. (5.5)
For i = 1, . . . , r, denote by k[i, 1] < · · · < k[i, Miθ] the elements of Ii,θ, set k[i, 0] = 0, and for j = 0, . . . , Miθ− 1:
(
∆ti,j := tk[i,j+1]− tk[i,j],
bE i,j := −θ(h2k[i,j+1]δ g k[i,j+1]− h 2 k[i,j]δ g k[i,j])/∆ti,j. (5.6) Set Gk(vk, zk) := g0(yk+1) − g0(yk) hk zk+ g0(yk+1) (f0(uk, yk)(vk, zk) − θhkδ y k) . (5.7) If z = zθ[v], then Gk(vk, zk) = g0(yk+1)zk+1− g0(yk)zk hk . (5.8)
Since z0= 0, it follows that:
g0(yθk)zk= k−1
X
q=0
hqGq(vq, zq). (5.9)
So the solution of (QP ) satisfies the following equality constraints, denoting by Gi,k
the i-th component of Gk: k[i,j]−1 X q=0 hqGi,q(vq, zq) = −θh2kδ g i,k[i,j], i = 1, . . . , r, j = 1, . . . M θ i − 1. (5.10)
An equivalent set of equality constraints is
bEi,j− 1 ∆ti,j k[i,j+1]−1 X k=k[i,j] hkGi,k(vk, zk) = 0, i = 1, . . . , r, j = 0, . . . Miθ− 1. (5.11)
The corresponding quadratic problem is (QPE) Min v 1 2Ω θ(v, z) − θ N −1 X k=0 h2k(δkpzk+ δukvk) s.t. z = zθ[v] and (5.11). (5.12)
We call vθ the solution of this problem, and δ ¯ηθ the multiplier associated with con-straints (5.11) and ˆq the costate. The Lagrangian of (QPE) is by the definition
Ωθ(v, z) + N −1 X k=0 ˆ qk+1(hkfk0(vk, zk) + zk− zk+1) +X i,j ∆ti,jδ ¯ηi,jθ bEi,j− k[i,j+1]−1 X k=k[i,j] hkGk(vk, zk)/∆ti,j . (5.13)
The optimality conditions of (QPE) have the following form. The costate equation is
ˆ qkθ= qˆk+1θ + hkqˆk+1θ f k y + hk(vkθ) THk uy+ hk(zθk) THk yy+ hkνkθ(z θ k) Tg00(y k) − r X i=1 δ ¯ηi,j[i,k]θ (g0(yk+1) − g0(yk) + hkg0(yk+1)fy(uk, yk)) − θh2kδ p k; ˆ qθ N = (zNθ)Tφ00(yNθ) + hNνNθ(zNθ)Tg00(yN). (5.14) Given k ≤ Mθ i, set
j[i, k] := min{j ∈ Ii,θ; j ≥ k + 1}. (5.15)
Expressing the stationarity of the Lagrangian w.r.t. v we get that
(vk)THuuk + (zk)THuyk + qˆ θ k+1− r X i=1 δ ¯ηi,j[i,k]θ gy(uk, yk) ! fuk= 0. (5.16)
This suggests to define
˜ qθ k+1 := ˆq θ k+1− r X i=1 δ ¯ηθi,j[i,k]g0(yk+1), k = 0, . . . , N − 1, δ ¯νθ k := δ ¯η θ i,j[i,k]− δ ¯η θ i,j[i,k−1], k = 0, . . . , N − 1, (5.17) Then ˜qθ is solution of ˜ qθ k= q˜θk+1+ hkq˜θk+1fyk+ hk(vkθ)THuyk + hk(zkθ)THyyk + hk(zkθ)Tg00(yk) +hkδ ¯νkθg 0 i(yk) − θh2kδ p k, ˜ qθ N = (z θ N) Tφ00(yθ N) + hNνNθ(z θ N) Tg00(yθ N) + hNδ ¯νNθg0(y θ N). (5.18)
In addition, we observe that if δ ¯νkθ 6= 0, then δ ¯ηθi,j[i,k] > δ ¯ηθi,j[i,k−1], and therefore the i-th state constraint is active at step k. Since (QP ) has a unique multiplier, we deduce that ˜qθ= qθ and δ ¯ηθ= δ ¯ηθi,j[i,k−1], and
5.2. Uniform surjectivity. We write here the linear mappings involved in the tangent quadratic problem, starting with
zk+1L = zLk + hkfy(uk, yk)(vk, zkL), k = 0, · · · , N − 1; z L
0 = 0. (5.20)
For any v in the space VN, (5.20) has a unique solution denoted by zL[v]. Let ξ k be solution of ξ0= 0 and ξk+1= ξk+ hkfykξk− θh2kδ k y; k = 0, . . . , N − 1; ξ0= 0. (5.21) We have that zk= zkL+ ξk; k = 0, . . . , N, kξk∞= O(¯h). (5.22) We set GL
i,j(v) := (gi0(yk[i,j+1])zk[i,j+1]L [v] − g 0
i(yk[i,j])zLk[i,j][v])/∆ti,j,
j = 0, . . . , Mθ
i − 1; i = 1, . . . , r.
(5.23) The linear (homogeneous) equations corresponding to those of (QPE) are therefore
GL(v) = 0. Consider the following perturbation of the r.h.s. of these equations, where
¯b is an arbitrary r.h.s.:
GL(v) = ¯b. (5.24)
We use the following norm, for s ∈ [1, ∞):
k¯bks s:= r X i=1 Mθ i−1 X j=0 ∆ti,j|¯bi,j|s. (5.25)
These norms can be identified with the usual Lsnorms on [0, T ] for piecewise constant functions and therefore we have the usual Cauchy-Schwarz and H¨older inequalities, in particular
k¯bk1≤
√
rT k¯bk∞ (5.26)
Proposition 5.2. There exist constants Cs, s ∈ [1, ∞], such that the linear
equation GL(v) = ¯b has, for small enough ¯h, a solution v verifying
kvks≤ Csk¯bks, for each s ∈ [1, ∞]. (5.27)
Before proving the Proposition 5.2 we introduce some notations. For t ∈ [0, T ], and ε0> 0, we define the set of ε0 active constraints as
Aε0(t) := {1 ≤ i ≤ r; |t
0− t| ≤ ε
0, for some t0 such that i ∈ A(t0)}. (5.28)
Since the control is continuous by (A1), and the first order state constraint satisfies (A2), we have that, for ε0> 0 small enough:
X i∈Aε(t) λi∇ug (1) i (¯u, ¯y) ≥1
2α|λ|, if λi= 0 when i 6∈ Aε0(t), for all t ∈ [0, T ].
For i ∈ {1, . . . , r} we denote the set of ε0active constraints by Jεi,θ0, i = 1, . . . , r. This
is a union of closed balls (in [0, T ]) of radius ε0. Since every connected component
has length at least 2ε0, Jεi,θ0 is a finite union of closed intervals.
We next define the one-time-step analogue of GL:
GLi,k(vk, zk) :=
gi0(yk+1) − g0i(yk)
hk
zk[v] + g0i(yk+1)(f0(uk, yk)(vk, zk[v])). (5.30)
Note that GLi,k(vk, zk) = (gi0(yk+1)zk+1[v] − g0i(yk)zk[v])/hk.
Proof. [Proof of Proposition 5.2] The idea is to compute, for each k, vk as the
minimum norm solution of the linear equations
GLi,k(vk, zkL) = ˜bi,k, i ∈ Aε(tk), (5.31)
where the variable size vector ˜bi,k, for i in Aε(tk), will be defined later, and then to
set
˜
bi,k:= GLi,k(vk, zKk ), for i 6∈ Aε(tk). (5.32)
Thanks to the expression of GL
i,k and (5.29) setting zL = zL[v], we have that
|vk| ≤ c1 |zL k| + X i∈Aε(tk) |˜bi,k| . (5.33)
Here the ci are positive constants not depending on v or k. It follows with (5.30) that
|˜bk| ≤ c2 |vk| + |zkL[v]| ≤ c3 |zkL| + X i∈Aε(tk) |˜bi,k| . (5.34) So, by (5.20): |zL k+1| ≤ (1 + c4hk)|zkL| + c5hk|vk| ≤ (1 + c6hk)|zkL| + c7hk X i∈Aε(tk) |˜bi,k|. (5.35)
By the discrete Gronwall’s Lemma it follows that
kzLk∞≤ c8 N −1 X k=0 hk X i∈Aε(tk) |˜bi,k|, (5.36)
and therefore, with and (5.33):
kvks s≤ c9 N −1 X k=0 hk X i∈Aε(tk) |˜bi,k|s, (5.37)
and in the r.h.s. we recognize the Lsnorm of the ε0active components of ˜b. We next
end the proof by fixing the ˜bk in such a way that
GL
We will obtain the second relation by induction over k, i.e. we will prove that there exists c > 0 such that
X `≤k |˜b`|s≤ c X i;k[i,j]≤k |¯bi,k|s. (5.39)
We distinguish two cases.
a) If {tk[i,j], . . . , tk[i,j+1]} ∈ Jεi,θ0 , then take
˜bi,k = ¯b
k[i,j+1], k = k[i, j] + 1, . . . , k[i, j + 1]. (5.40)
b) If {tk[i,j], tk[i,j+1]} 6∈ Jεi,θ0 , let k
0 be the smallest index in k[i, j], . . . , k[i, j + 1] such
that k ∈ Ji,θ ε0, whenever k 0 ≤ k ≤ k[i, j + 1]; Then tk0+1 2ε0≤ tk[i,j+1]. (5.41) We choose ˜bi,k = ¯ bi,j, k = k[i, j] + 1, . . . , k0, γ, k = k0+ 1, . . . , k[i, j + 1], (5.42)
for some γ such that
k0
X
k=k[i,j]+1
hk˜bi,k+ γ(tk[i,j+1]− tk0) = (tk[i,j+1]− tk[i,j])¯bk[i,j+1], (5.43)
so that the first relation in (5.38) holds. At the same time, since 1
2ε0≤ tk[i,j+1]− tk0, we have that γ ≤ 2 ε0
(tk[i,j+1]− tk[i,j])¯bk[i,j+1]− k0 X k=k[i,j]+1 hk˜bi,k ≤ 2 ε0 T |¯bk[i,j+1]| + X k≤k0 hk|˜bi,k| , (5.44)
and we use the induction hypothesis (5.39) in order to estimateP
k≤k0hk|˜bi,k|. The
conclusion follows.
From the same argument of the previous result we can deduce also an estimate for the control of a feasible trajectory.
Note that
Gi,k(vk, zk) = GLi,k(vk) + O(¯h), i = 1, ..., r. (5.45)
Corollary 5.3. Problem (QPE) has a feasible point ˜v, such that k˜vk∞= O(¯h).
Proof. In view of the Proposition above, it is enough to check that we can write the active constraints of (QPE) in the form
GL
i,k(v) = ¯bi,k; k¯bk∞= O(¯h). (5.46)
Remember that z[v] = zL[v] + ξk[v], with kξ[v]k∞= O(kvk1), and therefore
¯
bi,j= −θ
gi0(yk[i,j+1]) − g0i(yk[i,j])
∆ti,j
+ gi0(yk[i,j+1])fy0(uk[i,j], yk[i,j])
ξk[i,j] −θ(h2 k+1δ g k+1− h 2 kδ g k)/hk, (5.47)
which is of the desired form.
We recall a classical consequence of the coercivity of the cost function of an equality constrained quadratic problem over its feasible set. To keep the notation as simple as it possible, we formulate the problem in an abstract way. The result will be stated with the current notation as corollary (Corollary 5.6). Given two Hilbert spaces X and Y , identified with their dual, consider the optimization problem
Min
x∈X(c, x)X+
1
2(Hx, x)X subject to Ax = b in T , (5.48)
where (·, ·)X denotes the scalar product in X (and k · k := (·, ·)X) with a similar
convention for Y , H : X → X is symmetric, A ∈ L(X, Y ), c ∈ X and b ∈ Y . The Lagrangian of the problem is
(c, x)X+
1
2(Hx, x)X+ (λ, Ax − b)Y. (5.49)
The associated optimality conditions are
c + Hx + ATλ = 0; Ax = b. (5.50)
Lemma 5.4. Let α > 0 and cA> 0 be such that
(i) Coercivity: αkxk2≤ (Hx, x)X, for all x ∈ Ker A,
(ii) Strong surjectivity: For any b0 ∈ Y , there exists x0 ∈ X such that Ax = b0
and kx0k ≤ cAkb0k.
Then there exists κ > 0, function of α and cA, such that (5.48) has a unique solution
¯
x and associated Lagrange multiplier λ such that
k¯xk + kλk ≤ κ(kbk + kck). (5.51)
Proof. That (5.48) has a unique solution ¯x is an easy consequence of the coercivity (which in the case of a quadratic cost implies the strong convexity over the feasible set since the latter is a vector subspace). The uniqueness of the Lagrange multiplier λ is consequence of the surjectivity of A, implied by the strong surjectivity.
The latter also implies the existence of x0 such that Ax0= b and kx0|| ≤ c Akbk.
Then δx := ¯x − x0 is solution of
Hδx + ATλ = −c − Hx0; Aδx = 0. (5.52)
Therefore, since δx ∈ Ker A:
αkδxk2≤ δxTHδx = −(δx, c + Hx0) X+ (Aδx, λ), so that kδxk ≤ (kck + kHx0k)/α. Since kx0k ≤ c Akbk, we deduce that k¯xk ≤ kδxk + kx0k ≤ cAkbk + 1 α(kck + kHk cAkbk) . (5.53) By the surjectivity hypothesis
kλk ≤ 1 cA
kATλk ≤ 1
cA
(kck + kHkkxk), (5.54)
Given v ∈ VN, we denote by ¯v the associated corresponding piecewise constant
element defined by ¯
v(t) = vk, t ∈ (tk, tk+1), for all k = 0 to N − 1. (5.55)
Note that v and ¯v have the same Lsnorm, s ∈ [1, ∞] norm. Setting ¯z := z[¯v] (solution
of the linearized state equation (2.8) for the original (continuous time) problem, we easily check that
kzθ− ¯zk
∞= kvk∞O(ε + ¯h). (5.56)
We apply the previous result to (QPE) using the following Lemma:
Lemma 5.5. We have that for any v in UN:
Ωθ(v, zL[v]) − Ω(¯v)= O(εk¯vk)2.
Proof. Since by the definition (see (2.22)), hθkνkθ= ¯ηkθ− ¯ηk+1θ it follows that:
∆ := N X k=0 hkνkθD2gθk(zk)2= N X k=0 (¯ηθk− ¯ηθk+1)D2gθk(zk)2 = N X k=1 ¯ ηkθ(D2gkθ(zk)2− D2gθk−1(zk−1)2) = ∆1+ ∆2,
where by a Taylor expansion of D2gθ, for some ˆyk ∈ [yk−1, yk]:
∆1 := N X k=1 ¯ ηk(D2gkθ(zk)2− D2gk−1θ (zk)2) = N X k=1 hkη¯kD3gθ(ˆyk)(fk−1θ , zk, zk),
using the identity A(b, b) − A(a, a) = A(a + b, a − b) for any symmetric bilinear form A, and the linearized state equation
∆2 := N X k=1 ¯ ηkθ(D2gθk−1(zk)2) − D2gθk−1(zk−1)2) = N X k=1 hk−1ηkθD 2gθ k−1(zk+ zk−1, Dfk−1θ (vk−1, zk−1)).
We deduce that, for ¯h small enough, ∆1= Z T 0 g(3)(¯y(t))(f [t], ¯z(t), ¯z(t))¯η(t)dt + O(εk¯vk) and ∆2= 2 Z T 0 g00(¯y(t))(¯z(t), fy[t](¯z(t), ¯v(t))¯η(t)dt + O(εk¯vk).
Using the identity (2.19) we can claim that Ω
θ(v, z) − ˜Ω(¯v)
= O(εk¯vk)
2. We
con-clude with Lemma 2.4.
Corollary 5.6. Let assumptions (A0)-(A3) hold as well as either (A4) or (A5). We have that (QPE) has a unique solution vθ associated with a unique
alter-native multiplier δ ¯ηθ, and they satisfy kvθk2
2+ kδ ¯η θk2
Proof. The surjectivity of the constraint condition was proved in Proposition 5.2. If (A4) holds, then the coercivity property is an easy consequence of the stability after discretization of the Hessian of the Lagrangian (Lemma 5.5 above) and of the solution of the linearized state equation. If (A5) holds, the coercivity property is derived in Appendix A.
5.3. Estimate of the directional derivative. We arrive, finally, to the main result of the Section. We recall that δXθ:= (vθ, zθ, qθ, δηθ) is the directional
deriva-tive (on the left) of Xθ:= (uθ, yθ, pθ, ηθ).
Proposition 5.7. We have, for a fixed C > 0 kvθk
∞+ kzθk∞+ kqθk∞+ kδηθk∞≤ C¯h.
Proof. (a) Applying Lemma 5.4 to (QPE), where X and Y have norms defined
by (5.27) (where s = 2) and (5.25), having in mind that, by the definition of the Lagrangian, we have identified the image space (of bE) with its dual, we get that that
kvθk 2+ X i,j ∆ti,j|δ ¯ηi,jθ | 2≤ c 1¯h. (5.58)
Fix εη> 0, not depending on ¯h. Then
If ∆ti,j > εη, then |δ ¯ηθi,j| ≤ c1ε−1/2η ¯h. (5.59)
So, as far as δ ¯ηθis concerned, it remains to obtain a uniform estimate when ∆ti,j≤ εη.
It easily follows from (5.58), the linearized state equation (5.2), and the linearized costate equations (5.14) that
kzθk
∞+ kqθk∞≤ c2¯h. (5.60)
(b) By (5.6), |¯bi,j| = O(¯h). Evaluating the contribution of the term containing zk,
observingPk[i,j+1]
k=k[i,j]+1hk = ∆ti,j and then
1 ∆ti,j ki+1 X k=ki+1 hk∇uˆg (1) i,kvk= O(¯h). (5.61) Eliminating vk in (5.16), we get 1 ∆ti,j k[i,j+1] X k=k[i,j]+1 hk∇uˆg (1) i,k(H k uu) −1 r X i0=1 (∇ugˆ (1) i0,k)Tδ ¯ηθi0,j[i0,k]= O(¯h). (5.62) Multiplying by δ ¯ηθ
i,j[i,k] on the left, summing over i and j ∈ Ii0,k and setting
wk := r X i=1 (∇ugˆ (1) i,k) Tδ ¯ηθ i,j[i,k] (5.63) we get 1 ∆ti,j r X i=1 k[i,j+1] X k=k[i,j]+1 hkwTk(H k uu) −1w k = O(¯h)|δ ¯ηθ|. (5.64)
Table 6.1
Experimental error (DO NEW TESTS).
h kyh− ¯yk ∞ Ord(L∞) kuh− ¯uk∞ Ord(L∞) 0.05 0.2909 1.1704 0.025 0.16131 0.8506 0.6254 0.8776 0.0125 0.08313 0.9564 0.3304 0.9205 0.0063 0.04125 1.0109 0.1654 0.9982 0.0031 0.02059 1.0024 0.0829 0.9965 0.0016 0.01022 1.0105 0.0441 0.9106 0.0008 0.005187 0.9784 0.0222 0.9902 0.0004 0.002285 1.1827 0.0101 1.1362
Denote by ˆη the vector of components δ ¯ηθ
i,j[i,k], for i = 1 to r. For small enough ε
depending on ˆη, by (A2)-(A3), we have that |ˆη|2≤ α2g|wk|2≤ α−1α2gw
T k(H
k
uu)−1wk (5.65)
and therefore, since ∆ti,j =P k[i,j+1] k=k[i,j]+1hk: |ˆη|2≤ α −1α2 g ∆ti,j k[i,j+1] X k=k[i,j]+1 wkT(Huuk )−1wk= O(¯h)|ˆη|. (5.66)
Therefore, we get with (5.19) that
|δ ¯ηi,j[i,k]θ | ≤ O(¯h). (5.67) The corresponding estimate for vk and δηθ follow from (5.16) and (5.19).
6. Example. We present next an academic example which is a variant from the one in [16]. Let us consider the following optimal control problem, for some ε > 0: where g := 9.8:
Min R01 12u2(t) + gy(t) dt + (y(1) − 1)2/¯ε, s.t. ˙y(t) = u(t), y(0) = 1, y(t) ≥ 0,
The solution of this problem can be seen as the minimum energy state of a system composed by a rope of uniformly distributed mass in a constant gravity field, with the presence of a lower constraint (for example a table). We can add a state variable say ˜
y, with zero initial condition and derivative equal to the integrand of the integral cost, and reformulate the cost as ˜y(1) + (y(1) − 1)2/¯ε, in order to comply with the format
of the theoretical results. It is known that the costate associated with ˜y has value 1 (observe that this problem is qualified) and that the costate associated with y and the measure associated with the state constraints are invariant under this reformulation. We solved the discrete solution using a shooting method; in Figure 6 and Table 6 are shown the results at various constant discrete steps h and for ¯ε = 10−8. This test confirms the convergence results stated before.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.5 −0.45 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0
Fig. 6.1. Test with various discrete steps.
As shown before, a key point of the theory is the coercivity of the (QP ). Under the assumptions (A5), we can obtain it directly showing the stability of the boundary arcs. This will be sufficient to guarantee the coercivity of the Hessian of the Lagrangian. A main point is contained in the following Lemma:
Lemma A.1. Let (A0) − (A3) and (A5) hold. Given a boundary arc (ta, tb), let
ka and kb be defined as
ka (resp. kb): first index (resp. last index) for which tk > ta (resp. tk< tb). (A.1)
Let ε0 > 0. Reducing ε > 0 small enough, we have that when ¯h is small enough, the following holds:
g(yθk) = 0, for all 0 ≤ k ≤ N such that tka+ ε
0< t
k < tkb− ε
0.
Proof. (a) By the definition kηθ− ¯ηk
∞ < ε, and by (A5), ¯η has a uniformly
positive derivative over (tka+ ε
0< t
k < tkb− ε
0) minored by c
1> 0. We have that for
ka< k < kb: ηθk≥ ¯η(tk) − ε ≥ ¯η(tka) + c1(tk− tka) − ε ≥ η θ ka+ c1(tk− tka) − 2ε, (A.2) that is, c1(tk− tka) − 2ε ≤ η θ k− η θ ka. (A.3)
Therefore, if tk− tka > 2ε/c1, the above r.h.s. must be positive, proving that the
constraint is active for some k such that tk≤ tka+ 2ε/c1.
(b) If the conclusion does not hold, by step (a), it suffices to prove that g(yθ
k) cannot
have a negative local minimum for some ka < k < kb. We give a proof by
contradic-tion. If this was the case, then ∆k
g ≥ 0, defined in (4.6), and νk = 0. Multiplying
(4.3) by ∇ug(1)(uk, yk)T(Hθk)−1 on the left and using Lemma 4.1(ii) and (4.8), we get
−∆
k g
hk
where ˆΞθ
k is such that kˆΞθ− ˆΞ1k∞= O(ε). We have that the l.h.s. of (A.4) is greater
than a positive constant independent on ¯h, since, for θ = 1, for some K > 0, ∆k g= 0, νk > K and |∇g (1) u | > K, i.e., −∆ k g hk + νk(∇ug(1)(uk, yk))TH−1k ∇ug(1)(uk, yk) ≥ C. (A.5)
This relation is still valid for ε and ¯h small enough, for all θ ∈ [θm, 1], in view of
the continuity of the r.h.s. of (A.4). However, if a negative minimum of the state constraint is attained at index k, then νk= 0 and ∆kg≥ 0, contradicting (A.5).
Here we prove the coercivity of Ωθ over the feasible domain of (QP ). Note that
such a property is naturally preserved passing to the alternative formulation ˜Ωθ as
shown, for the continuous case in Section 2.
Lemma A.2. Let (A5) hold. Then v 7→ Ωθ(v, ZL[v] is uniformly (over ¯h small enough) coercive over the feasible domain of (QP ).
Proof. (a) We first examine the continuous problem and prove that Ω is, for ε > 0 small enough, coercive over the following enlargement of the critical cone:
Cε:= {v ∈ V | gk = 0, for all 0 ≤ k ≤ N such that tka+ ε < tk < tkb− ε}. (A.6)
Indeed, otherwise we would have a sequence εq ↓ 0 and vq in Cεq such that Ω(v
q) ≤
o(1). Extracting if necessary a subsequence, assume that vq weakly converges to ¯v in
V. Thanks to the Legendre condition we have that Ω is a Legendre form and therefore
Ω(¯v) ≤ lim inf
q→0 Ω(v
q) ≤ 0. (A.7)
At the same time; by standard compactness arguments g0(¯y)¯z = 0 when the constraint is active (where ¯z is the linearized state associated with ¯v), and so, ¯v is a critical direction. So, Ω(¯v) ≤ 0 implies that ¯v = 0. But then Ω(¯v) = limqΩ(vq), so that vq
(of unit norm) strongly converges to ¯v, which gives the desired contradiction. (b) Now let v belong to the feasible domain of (QP ). By Lemma A.1, we know that v belongs to the set
{v ∈ VN; |g0(yθ
k)zk| ≤ ε, 0 ≤ k ≤ N such that tka+ ε < tk< tkb− ε.}. (A.8)
Let ¯v be the associated element of V and ¯z the corresponding linearized state. Given ε > 0, it is easily checked that ¯v ∈ Cε when ¯h is small enough, and so, by step (a),
Ω(¯v) ≥ 12αk¯vk2. We conclude with Lemma 5.5.
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